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The Uniform Electron Gas Pierre-François Loos* and Peter M Advanced Review The uniform electron gas Pierre-François Loos* and Peter M. W. Gill The uniform electron gas or UEG (also known as jellium) is one of the most fundamental models in condensed-matter physics and the cornerstone of the most popular approximation—the local-density approximation—within density- functional theory. In this article, we provide a detailed review on the energetics of the UEG at high, intermediate, and low densities, and in one, two, and three dimensions. We also report the best quantum Monte Carlo and symmetry- broken Hartree-Fock calculations available in the literature for the UEG and discuss the phase diagrams of jellium. © 2016 John Wiley & Sons, Ltd How to cite this article: WIREs Comput Mol Sci 2016, 6:410–429. doi: 10.1002/wcms.1257 INTRODUCTION density correlation functionals (VWN,8 PZ,9 PW92,10 etc.), each of which requires information fi he nal decades of the 20th century witnessed a on the high- and low-density regimes of the spin- Tmajor revolution in solid-state and molecular polarized UEG, and are parametrized using numer- physics, as the introduction of sophisticated 1 ical results from quantum Monte Carlo (QMC) exchange-correlation models propelled density- calculations,11,12 together with analytic perturbative functional theory (DFT) from qualitative to quantita- results. tive usefulness. The apotheosis of this development For this reason, a detailed and accurate under- was probably the award of the 1998 Nobel Prize for 2 3 standing of the properties of the UEG ground state is Chemistry to Walter Kohn and John Pople but its essential to underpin the continued evolution of origins can be traced to the prescient efforts by Tho- DFT. Moreover, meaningful comparisons between mas, Fermi, and Dirac, more than 70 years earlier, to theoretical calculations on the UEG and realistic sys- understand the behavior of ensembles of electrons tems (such as sodium) have also been performed without explicitly constructing their full wave recently (see, e.g., Ref 13). The two-dimensional functions. (2D) version of the UEG has also been the object of In principle, the cornerstone of modern DFT extensive research14,15 because of its intimate connec- – 4 is the Hohenberg Kohn theorem but, in practice, tion to 2D or quasi-2D materials, such as quantum it rests largely on the presumed similarity between dots.16,17 The one-dimensional (1D) UEG has the electronic behavior in a real system and that recently attracted much attention due to its experi- – in the hypothetical three-dimensional (3D) uniform mental realization in carbon nanotubes18 22 organic 5 – electron gas (UEG). This model system was conductors,23 27 transition metal oxides,28 edge applied by Sommerfeld in the early days of quan- 29–31 6 states in quantum Hall liquids, semiconductor tum mechanics to study metals and in 1965, 32–36 fi 37–39 7 heterostructures, con ned atomic gases, Kohn and Sham showed that the knowledge of a and atomic or semiconducting nanowires.40,41 In the analytical parametrization of the UEG correlation present work, we have attempted to collect and col- energy allows one to perform approximate calcula- late the key results on the energetics of the UEG, tions for atoms, molecules, and solids. This information that is widely scattered throughout the spurred the development of a wide variety of spin- physics and chemistry literature. The UEG Paradigm section defines and describes the UEG model in *Correspondence to: [email protected] detail. The High-Density Regime section reports the Research School of Chemistry, Australian National University, known results for the high-density regime, wherein Canberra, Australia the UEG is a Fermi fluid (FF) of delocalized elec- Conflict of interest: The authors have declared no conflicts of inter- trons. The Low-Density Regime section reports est for this article. analogous results for the low-density regime, in 410 ©2016JohnWiley&Sons,Ltd Volume6,July/August2016 WIREs Computational Molecular Science The uniform electron gas ð which the UEG becomes a Wigner crystal (WC) of ρðÞr vðÞr dr + J½ρ + E =0; ð6Þ relatively localized electrons. The intermediate- b density results from QMC and symmetry-broken Hartree–Fock (SBHF) calculations are gathered in which yields the Intermediate-Density Regime section. Atomic units are used throughout. E½ρ = Ts½ρ + Exc½ρ = T ½ρ + E ½ρ + E ½ρ ð s x ð c ð ð7Þ = ρe ½ρ dr + ρe ½ρ dr + ρe ½ρ dr: UEG PARADIGM t x c The D-dimensional UEG, or D-jellium, consists of In the following, we will focus on the three reduced interacting electrons in an infinite volume in the pres- (i.e., per electron) energies et, ex, and ec, and we will ence of a uniformly distributed background of posi- discuss these as functions of the Wigner–Seitz radius tive charge. Traditionally, the system is constructed rs defined via by allowing the number n = n" + n# of electrons 8 (where n" and n# are the numbers of spin-up and >4π <> r3, D =3, spin-down electrons, respectively) in a D-dimensional πD=2 s fi 1 ÀÁD 3 ð Þ cube of volume V to approach in nity with the den- = rs = π 2 8 1 ρ Γ D > rs , D =2, sity ρ = n/V held constant. The spin polarization is 2 +1 > : 2r , defined as s D =1, ρ −ρ − or ζ " # n" n# ; ð Þ = ρ = 1 8 n > 1=3 > 3 , D =3, > 4πρ > where ρ" and ρ# is the density of the spin-up and < 1=2 1 spin-down electrons, respectively, and the ζ = 0 and r = πρ , D =2, ð9Þ s > ζ = 1 cases are called paramagnetic and > 1 > , D =1, ferromagnetic UEGs. :> 2ρ The total ground-state energy of the UEG (including the positive background) is Γ 42 ð where is the Gamma function. It is also conven- ient to introduce the Fermi wave vector E½ρ = Ts½ρ + ρðÞr vðÞr dr + J½ρ + Exc½ρ + Eb; ð2Þ α k = ; ð10Þ F r where Ts is the noninteracting kinetic energy, s ð ρ ðÞr0 where vðÞr = − b dr0 ð3Þ jjr −r0 8ÀÁ 1=3 > 9π , D =3, >p4ffiffiffi 2=D < is the external potential due to the positive back- D−1 D 2, D =2, ρ α =2 D Γ +1 = π ð11Þ ground density b, 2 > > , D =1: ðð : 4 1 ρðÞr ρðÞr0 J½ρ = drdr0 ð4Þ 2 jjr−r0 is the Hartree energy, Exc is the exchange-correlation energy and THE HIGH-DENSITY REGIME In the high-density regime (r 1), also called the ðð 0 s 1 ρ ðÞr ρ ðÞr 0 weakly correlated regime, the kinetic energy of the E = b b drdr ð5Þ b 2 jjr −r0 electrons dominates the potential energy, resulting in a completely delocalized system.5 In this regime, the is the electrostatic self-energy of the positive back- one-electron orbitals are plane waves and the UEG is ground. The neutrality of the system [ρ(r)=ρb(r)] described as a FF. Perturbation theory yields the implies that energy expansion Volume6,July/August2016 ©2016JohnWiley&Sons,Ltd 411 The values of The values of and the spin-scaling function is where magnetic limits are given in Table 1 for exchange energy where the noninteracting kinetic energy where relation energy order perturbation energies, respectively, and the cor- magnetic limits are given in Table 1 for 412 (12), can be written The exchange energy,Exchange which Energy is the second term in and 3. mas and Fermi The 3D case hasfi been known since theThe work noninteracting of kinetic Tho- energyNoninteracting of kinetic energy orders. rst term of the high-density energy expansion (12). Advanced Review e FF ðÞ r s , ζ Υ Υ ε x x t ε = 43,44 ðÞ ðÞ ≡ t ε ε ζ ≡ ζ e x t ε t ( ( ε ðÞ x ζ e ζ = e = t r ðÞ e ) in the paramagnetic and ferro- c FF ðÞ ε ) in the paramagnetic and ferro- x e s 47,48 ε ζ and, for x ζ ðÞ , x ( ðÞ t t ðÞ r ðÞ ðÞ 1+ ζ 1+ ðÞ =0 ðÞ =0 r r s r ζ ζ , s s s + , , , ζ ζ ζ ζ ζ = ζ = ) are the zeroth- and e = = x ε D ε D = = ðÞ D x − t D +2 +1 2 Υ r Υ encompasses all higher D ε ε s ðÞ π 2 +1 2 t +1 , x t x D r ðÞ r ðÞ -jellium, it reads ðÞ ζ ðÞ ðÞ D s 2 ζ ðÞ s ζ D ðÞ ζ ζ 2 +2 + ; 2 D ; − ; − ; − e ζ ζ c FF 1 α D D 2 ðÞ D D D +2 α +1 r ; s ; -jellium is the D : , : ζ e t =1,2 ©2016JohnWiley&Sons,Ltd Volume6,July/August2016 ; ( D r s , 45,46 =1,2 ζ ð ð ð ð ) and ð , 14b 17b 17a 14a fi 17c ð ð ð ð and 15 16 12 13 rst- Þ Þ Þ Þ Þ Þ Þ Þ Þ , TABLE 1 | Energy Coefficients for the Paramagnetic (ζ = 0) and Ferromagnetic (ζ = 1) States and Spin-Scaling Functions of D-jellium at High Density Paramagnetic State Ferromagnetic State Spin-Scaling Function ε(0), λ(0) ε(1), λ(1) Υ(ζ), Λ(ζ) Term Coefficient D =1 D =2 D =3 D =1 D =2 D =3 D =1 D =2 D =3 − 2 ÀÁ= 2 ÀÁ= 2 ε ζ π 3 9π 2 3 π 2=3 3 9π 2 3 rs t( ) /96 1/2 /24 1 2 1 Eq. (15) Eq. (15) pffiffi 10 4ÀÁ 10 4ÀÁ −1 ε ζ −∞ 4 2 π 1=3 −∞ 8 1=3 π 1=3 r x( ) − − 3 9 − π − 3 9 1 Eq.
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